Question

Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about $1^{\circ} \mathrm{C}$ for each 100 -meter rise, up to about 12 $\mathrm{km}.$ (a) If the ground temperature is $20^{\circ} \mathrm{C},$ write a formula for the temperature at height $h .$ (b) What range of temperatures can be expected if an airplane takes off and reaches a maximum height of 5 $\mathrm{km}$ ?

Name: Problem 84, Chapter 1: Algebra and Trigonometry
Uploaded: 2019-06-20T19:38:51-08:00
Duration: 3 min 48 s
Channel: Ernest Castorena
Description: Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about 1^∘ C for each 100 -meter rise, up to about 12 km. (a) If the ground temperature is 20^∘ C, write a formula for the temperature at height h . (b) What range of temperatures can be expected if an airplane takes off and reaches a maximum height of 5 km ?

   Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about $1^{\circ} \mathrm{C}$ for each 100 -meter rise, up to about 12 $\mathrm{km}.$
(a) If the ground temperature is $20^{\circ} \mathrm{C},$ write a formula for the temperature at height $h .$
(b) What range of temperatures can be expected if an airplane takes off and reaches a maximum height of 5 $\mathrm{km}$ ?

Algebra and Trigonometry

James Stewart,… 4th Edition

Chapter 1, Problem 84

Instant Answer

Solved by Expert Ernest Castorena

06/20/2019

Step 1

We are also given that the ground temperature is $20^{\circ} \mathrm{C}$. We can use these two pieces of information to write a formula for the temperature at height $h$. The formula is $x=20^{\circ}-\frac{h}{100}$. Show more…

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Air Temperature As dry air moves upward, it expands and, in so doing, cools at a rate of about $1^{\circ} \mathrm{C}$ for each 100 -meter rise, up to about 12 $\mathrm{km}.$ (a) If the ground temperature is $20^{\circ} \mathrm{C},$ write a formula for the temperature at height $h .$ (b) What range of temperatures can be expected if an airplane takes off and reaches a maximum height of 5 $\mathrm{km}$ ?

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Key Concepts

Temperature Range Calculation

Temperature range calculation involves determining the span between the maximum and minimum temperatures over a certain interval. In problems like these, it is about evaluating a linear temperature model at different altitudes to find the expected temperature variation.

Environmental Lapse Rate

The environmental lapse rate describes the rate at which temperature decreases with an increase in altitude. This concept is vital in meteorology and atmospheric science, as it explains the cooling of air as it rises, typically due to expansion and adiabatic cooling effects.

Linear Functions

Linear functions represent relationships where the change between two variables is constant, typically expressed in the form y = mx + b. They are fundamental in modeling equations where a variable, such as temperature, changes uniformly with another variable, like altitude.

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